Counting Number of Card Deck Orderings to fit Rules of Game

Question

I am investigating an interesting problem regarding a simple card game, and I ultimately want to calculate the probability that this game results in all the cards in a hand end up with the same value (they change based on the game rules).

In this game, all of the "cards in hand" of any number of players eventually converge to the same card after a certain number of iterations of the game. This game proceeds as follows:

1) Each player randomly selects a card from a standard shuffled deck. These cards are known as the ”cards in hand”, or ”hand cards”. Once recorded, these cards are reinserted into the deck, which is subsequently reshuffled.

2) A card is drawn from the top of the deck. If the suit of the drawn card matches the suit of a player’s card in hand, the player then changes their recorded card in hand to that of the card subsequent the drawn card. For example, if a player’s current card is a Jack of Spades, the drawn card is an Ace of Spades, and the next card in the deck is a Queen of Hearts, the player’s card in hand changes to a Queen of Hearts. This process constitutes one iteration. The card is not reinserted into the deck, and the deck is not reshuffled. Then, another iteration begins.

3) At the end of the game, all of the recorded cards in hand should be the same.

In the version of the game I am investigating, I assume that there are three players, i.e. three "hand cards", and that they all pick cards of different suits. What I need help finding is the number of orderings of the deck (from which cards are drawn) so that at the end when all cards are draw, the three cards still have different suits (not necessarily the same ones). I was able to deduce, based on the game rules, that if the three cards have different suits in the hand, then they MUST have different suits at any point during the duration of this game. If this is to happen, then for every card draw, we can have:

a. the card drawn from the deck matches the suit of one of the hand cards, and the subsequent card drawn from the deck has the same suit as both - in this case the suits do not change in the hand

b. the card drawn from the deck matches the suit of one of the hand cards, and the subsequent drawn card from the deck is of the one suit that is NOT present in the hand (Spade, Heart, Club goes to Diamond, Heart, Club)

c. the card drawn from the deck does not match the suit of any of the hand cards, in which case nothing happens and the next card is drawn

I know this is a hard and lengthy question, and I will be grateful to whoever can help me figure it out.

Answer 1

Assuming my understanding of your statements is correct, the key thing to note here is that once your suit shows up in the draws, your card will change with every subsequent draw. If draw 1 is your suit, then your new card-in-hand is the card for the 2nd draw. but in that case, the 2nd draw is your suit as well, so your card switches to that of the 3rd draw and so on.

Since every suit occurs in the deck, as soon as the 4th suit appears in a draw, everybody is switching cards every time, and so they will always end with the final card in the deck.

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If instead, you discard 2 cards with each turn - the first being the "draw", and the second being the new card-in-hand for all matching players, then things are a little more complicated, but not by much. The first time a draw of one suit comes with a card-in-hand from another suit, all players will from that point on be in only 3 suits. The players in the first suit will join those in the second. Even if players transfer back to that original suit, everyone in the transfering suit will do so, leaving that suit with no players. When a third suit comes up as card-in-hand for a different suit, then another suit will be lost, and all players will be in just 2 suits from then on. Eventually it is likely that one of the 2 suits will end up transferring to the other. After that, only one suit will remain. And if that suit ever shows up in the draw, everyone will end up with the same card.

Since each suit has 13 cards, it is not possible for them to show up paired only with a card-in-hand of the same suit. Therefore, at least one suit will be lost. However, it is possible to limit that loss to just one suit, so at the end of the game, there are players in each of the other three suits. For example, The first three pairs could be: $$\text{heart} \to \text{spade}\\ \text{heart} \to \text{club}\\ \text{heart}\to \text{diamond}$$ which removes an odd number of cards from each suit. If the remaining pairs of cards has the new card-in-hand from the same suit as the draw, only those players starting with a heart will move to another suit: spades. Spades, clubs, and diamonds will remain in those suits to the end. Though everybody in each suit will see their card-in-hand switch to the same value, there will still be 3 different card-in-hands at the end.